4.) Nuclear structure (Lilley Chap. 2)

4.)

Nuclear structure (Lilley Chap. 2)

Models

Nuclear force

This is a short range attractive force, but repulsive for even shorter distances⇒ There is a certain optimal distance between nuclear particles.

Liquid drop model

The nucleus is considered as a spherical liquid drop with constant internal density.

Evidence for the existence of the liquid drop model:

The internal charge distribution:

a.) Electron scattering experiments imply the charge density function below:

Number of nucleons per unit volume is approximately constant⇒ ρ= 4^A 3πR³

b.) The nuclear charge distribution affects the energy levels of the S-orbital electrons.

c.) The potential energy difference between mirror nuclei:

Example:

137 N6 β⁺

→¹³₆ C7, Measure Emaxforβ⁺

∆EC= ³_54πε^e2

0

1 R

hZ²−(Z−1)²i

| {z }

(2Z−1)=A

⇒∆Ec= ³_54πε^e2

0

1 R0A²³

The internal mass distribution:

a.) Neutron scattering (elastic)

This is the same calculation as used for electron scattering, remembering to exchange the electron’s electro-magnetic potential with the neutron’s potential

⇒Scattering data give the Fourier transform of the mass distribution.

b.) Deviation from the expected angular dependency of Rutherford scattering forr > R.

c.) Calculating the tunneling probability forα-disintegration.

d.) Measuring the difference between Ek-energies for atoms with π−mesons

| {z } Strong force+Coulomb

and muons

| {z } Coulomb only

instead of electrons.

These four points, from a.) through d.), result in a conclusion:ρm'ρe,R=R0A¹³,R0= 1.2fm

Measuring atomic masses

Mass excess, ∆ =m−A is

≥0, ifA <12

≤0, ifA >12

Binding energy

Binding energy: B=h

Zm(¹H) +N mn−m(^A_ZX)i

c², wherec²= 931.5^{M eV}_u Neutron separation energy: Sn=h

mn+m(^A−1_Z X_N−1)−m(^A_ZXN)i c² Proton separation energy: Sp=h

mp+m(^A−1_Z−1XN)−m(^A_ZXN)i c²

Binding energy: B =av·A−asA²³ −ac·Z(Z−1)A⁻¹³ (Liquid drop model)

−asym·^(A−_A^2Z)2 +δpair (Shell effects)

Where δ=







+apA⁻³⁴, if Z & N are even numbers 0, if A is an odd number

−apA⁻³⁴, if Z & N are odd numbers

Semi-empirical mass formula: M(Z, A) =Zm(¹H) +N mn−^B(Z,A)_c2

M(A,Z) is sketched below for fixed values of A:

Minimum mass: ^∂M_∂Z = 0⇒Z=Zmin= ^{[mn−m(1H)]+acA−}

1 3+4a_sym 2acA⁻¹³+8asymA⁻¹

The nuclear shell model

This model is the nuclear analogy to the electron shell model.

Experimental data show that the ionisation energy decreases and the atomic radius increases rapidly for the first electron outside a full shell. I.e for Li, Na, K etc. The same occurs for nucleons in the nucleus.

Experimental data that justify the theory of a nuclear shell structure

a.) There is a rapid fall in 2-neutron and 2-proton separation energy when passing the magic nucleon numbers; 8, 20, 28, 50, 82, 126

b.) α-energy reaches maximum for radio-nuclei where the daughter nucleus has a structure corresponding to magic numbers.

c.) The neutron scattering cross-section for nuclei with N=magic numbers is extraordinarily small.

d.) There is a huge increase in the nuclear radius when the number of neutrons exceed magic numbers.

A realistic potential for the shell model (Woods-Saxon potential):

V = −V0

1 +e^r−Ra (1)

WhereV0'50M eV,R=R0A¹³,R0= 1.2fm

Spin-Orbit coupling

Energy difference: ∆E=−(~l·~s)Vso, Vso>0 Total angular momentum: ~j=~l+~s

From this, it follows that < ~l·~s >= ¹₂<[j~²−l~²−s~²]>=¹₂[j(j+ 1)−l(l+ 1)−s(s+ 1)]¯h² Energy splitting: δE=Vso[< ~l·~s >_j=l−¹

2 −< ~l·~s >_j=l+¹

2]=^¯h₂²Vso(2l+ 1)

# of identical nucleons per energy level: (2j+1)

Remember that the Pauli principle applies only for identical Fermions (protons and neutrons are counted independently).

Parity: (−1)^l ⇒

π⁺ for s, d, g..

π⁻ for p, f, h..

This shell model with spin-orbit coupling gives the right spin and parity. Further on, it predicts reasonable energy levels, and introduces the magical numbers corresponding to filled shells.

Angular momentum and spin

For each nucleon: ~j =~l+~s

For the nucleus: ~I=P~ji

~I²= ¯h²I(I+ 1) Iz=m¯h For nuclei with one valence-nucleon: I~=~jvn

For nuclei with two valence-nucleons: I~=~j1+~j2

For nuclei with even numbers of A: I∈integer For nuclei with odd numbers of A: I∈half integer

For even-even nuclei(Z&A even): I=0 in the ground state

Valence nucleons

Excited states: The valence nucleon jumps to a higher energy state in the shell model by absorbing excitation energy. This model agrees with experimental data for nuclei with one valence nucleon.

Experimental data which justify the orbital model for nucleons

Electron-scattering experiments to find the charge-distribution difference between ²⁰⁶₈₂ P b124 and

205

81 T l124. The difference, ∆ρe, takes place because Pb has one extra proton in a 3S¹

2-state.

⇒∆ρecorresponds to a 3s¹

2-orbital.

Protons and neutrons are found as proton- and neutron-pairs in the shell structure. To excite a nucleon, one has to break a pair bond (typically 2MeV binding energy). Energy and spin is then found from the two odd nucleons. Coupling of the two angular momenta~j1+~j2 gives values from

|j1+j2|to |j1−j2|.

Collective structure contributions in even-even nuclei

Experimentally:

All even-even nuclei have a low 2⁺ excited state with excitation energy around half the energy re- quired to separate a pair of nucleons, indicating another type of excited state than single nucleon excitation.

Experimental data:

Nuclear vibrations(for A < 150 )

The nuclear surface:

R(t) =Rav+X

λ≥1

Xλ µ=−λ

αλµ(t)Yλµ(θ, φ) (2)

A nuclear quadrupole-moment corresponds toY20(l= 2)

Exited phonon states with equidistant energy levels⇒E=n·¯hω

If the 4⁺ state is due to a two-phonon excitation and 2⁺ corresponds to a one-phonon excita- tion, one can easily draw the conclusion that E(4⁺)/E(2⁺) = 2. Experimental data forA < 150 confirms this model.

Rotating deformed nuclei ( 150 < A < 190 , A > 220 )

R(θ, φ) =R0

1 +βY20(θ, φ)

(3)

Deformation parameter: β =⁴₃p_π

5

∆R Rav ' _R^∆R

av

Intrinsic quadrupole moment, Q ,in the nucleus’ rest frame: Q0=^√3_5π ·R²_avZβ(1 + 0.16β)

A rotatingprolate

| {z }

Q0>0

ellipsoid rotates perpendicular to the symmetry-axis⇒Q <0.

Rotational states

E= ¯h²

2ΥI(I+ 1) (4)

The ground state for even-even nuclei has a total angular momentum I=0, and superimposed ro- tational states have even spin due to symmetry. Υ is the effective nuclear mass moment of inertia.

Deformed nuclei are found whereZ&N take values far from magic numbers.

Super-deformation

The Schrødinger equation for deformed nuclei gives a new set of states. When deforming a nucleus '2:1 prolate ellipsoid, a new shell structure arises⇒super-deformed states.